For many years now, employers have been required to have the new employee provide proof of lawful employability after the commencement of employment. A Green Card, an EAD (Work Card) an I-94 showing Asylum or Refugee status are just a few ways an alien can prove they can work lawfully. Anyone with a U.S. Drivers License and an Unrestricted Social Security card can successfully claim that they are authorized to work in the United States.
There have been rumors and information in the alien community for many years that one of the easy ways to work was to claim to be a U.S. Citizen and present a Social Security Card and a Drivers License. In years past, there have been may social security cards issued to aliens and many drivers licenses issued. I assume that many have done this to be able to work in the United States.
Since the 1990s, making a claim of U.S. Citizenship to gain an immigration benefit has been an offense that makes you Deportable and Inadmissible for life. There are many ways to be lawfully employed in the United States and the requirements are covered in the M-374 which is available on the USCIS web site.
If you are faced with producing an I-9 for USCIS or being denied a benefit, please consider talking to legal counsel before supplying it. But remember, in today’s environment, many employers will give out even the most private employee information to someone from USCIS on the telephone.
 Deportable under INA § 237(a)(3)(D).
 Inadmissible under INA § 212(a)(6)(C)(ii).
 Some narrow exceptions were added in 2000.